Rayleigh surface mode

Fig. 11.4. Velocities of bulk and surface waves in an (001) plane the angle of propagation in the plane is relative to a [100] direction, (a) Zirconia, anisotropy factor Aan = 0.36 (b) gallium arsenide, anisotropy factor Aan = 1.83 material constants taken from Table 11.3. Bulk polarizations L, longitudinal SV, shear vertical, polarized normal to the (001) plane SH, shear horizontal, polarized in the (001) plane. Surface modes R, Rayleigh, slower than any bulk wave in that propagation direction PS, pseudo-surface wave, faster than one polarization of bulk shear wave propagating in...

The surface Fuchs-Kliewer modes, like the Rayleigh modes, should be regarded as macroscopic vibrations, and may be predicted from the bulk elastic or dielectric properties of the solid with the imposition of a surface boundary condition. Their projection deep into the bulk makes them insensitive to changes in local surface structure, or the adsorption of molecules at the surface. True localised surface modes are those which depend on details of the lattice dynamics of near surface ions which may be modified by surface reconstruction, relaxation or adsorbate bonding at the surface. Relatively little has been reported on the measurement of such phonon modes, although they have been the subject of lattice dynamical calculations [61-67],... 

The harmonic-oscillator and elastic-continuum models can be used to explain the presence of surface phonons (Rayleigh waves and localized surface modes of vibration) and the larger mean-square displacement of surface atoms compared to that of atoms in the bulk. 

These three salts are isobaric because the masses of the anions and cations are nearly the same. Although crossing resonances were somewhat unexpectedly observed in the studies of KBr and to a lesser extent in Rbl, for these compounds an Sg mode, the folded extension of the Rayleigh wave as in Fig. 6, was anticipated [70]. For the isobaric case, this vibration appears to be a tme surface mode at least for part of the way across the SBZ and not just a resonance with a high density of states at the surface [70],... 

Phonon bands occur in the SBZ, similarly to the surface states discussed in Sect. 5.2.3. When the frequency of a surface mode corresponds to a gap in the bulk spectrum, the mode is localized at the surface and is called a surface phonon. If degeneracy with bulk modes exists, one speaks of surface resonances. Surface phonon modes are labeled Sj ( / = 1, 2, 3,...), and surface resonances by Rj when strong mixing with bulk modes is present, the phonon is labeled MSj. The lowest mode that is desired from the (bulk) acoustic band is often called the Rayleigh mode, after Lord Rayleigh, who first predicted (in 1887) the existence of surface modes at lower frequencies than in the bulk. 

Fig. 4.1. Phonon dispersion curves in MgO(lOO) (according to Chen et ai, 1977). Hatched zones are the projection of the bulk modes. Surface modes S are indexed by n (1 < n < 7) the Rayleigh mode is Si the Fuchs and Kliewer modes have a frequency close to 12x10 rad s S3 is an example of a microscopic mode.

As in the case of metals and semi-conductors, there exist specific surface excitations in insulating oxides. Three types of surface phonon modes may be distinguished the Rayleigh mode, the Fuchs and Kliewer modes and the microscopic surface modes. The first two modes have a long penetration length into the crystal. They are located below the bulk acoustic branches and in the optical modes, respectively. The latter are generally found in the gap of the bulk phonon spectrum. 

From each bulk band, at least one surface mode originates, as shown in Figure 9.46. According to Rayleigh s theorem [82], the number of localized surface modes from each bulk band is given by the number of degrees of freedom, which are affected by the surface perturbation. For an ideal bulk-terminated surface, as in the model calculation of Figure 9.46, no more than one surface phonon mode is expected for each bulk band. 

The mechanical breakup mode occurs around the rims of the sheet where the air-liquid relative velocity is low, forming relatively large droplets. At low relative velocities, aerodynamic forces are much smaller than surface tension and inertia forces. Thus, the breakup of the liquid rims is purely mechanical and follows the Rayleigh mechanism for liquid column/jet breakup. For the same air pressure, the droplets detached from the rims become smaller as the liquid flow rate is increased. 

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